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Ellipse is similar to the properties of a circle. There are several Ellipse Formula, like its area, perimeter, volume, circumference and more. Unlike the circle, an ellipse is oval in shape. Examples of the ellipse in our daily life are the shape of an egg in a two-dimensional form and the race track in a stadium. The ellipse is one of the conic sections, produced when a plane cuts the cone at an angle with the base. An ellipse has an eccentricity of less than one, and it also denotes the locus of points, the sum of whose distances from the two foci of the ellipse is often also called the constant value.
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Key Terms: Ellipse, Ellipse Formula, Fixed Points, Constant Value, Conic Sections, Foci, Locus Point
What is Ellipse?
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An ellipse is the locus of points in a plane, the sum of the distances from two fixed points (F1 and F2) is a constant value. The two fixed points (F1 and F2) are called the foci of the ellipse.
Ellipse
Ellipse is similar to other parts of the conic section like the parabola and hyperbola, which are open in shape and unbounded. A circle is an ellipse, where the foci are at the same point, which is the centre of the circle.
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Conic Sections Detailed Video Explanation:
Properties of Ellipse
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There are several properties of an ellipse, some include:
- Ellipse has two focal/fixed points, called foci.
- The fixed distance is called a directrix.
- The eccentricity of the ellipse lies between 0 to 1, 0 ≤ e < 1.
- The total sum of the distances from the locus of an ellipse to the two focal points is constant.
- Ellipse has a major axis, a minor axis and a center.
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Formula of Ellipse
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Some of the important formulas of Ellipse are:
Area of Ellipse Formula
Area of the Ellipse Formula = πr1r2
Where, r1 is the semi-major axis of the ellipse and r2 is the semi-minor axis of the ellipse.
Perimeter of Ellipse Formula
Perimeter of Ellipse Formula = 2 π√[(r21 + r22)/2]
Where, r1 is the semi-major axis of the ellipse and r2 is the semi-minor axis of the ellipse.
Ellipse Volume Formula
The volume of an elliptical sphere can be calculated with a simple and elegant ellipsoid equation:
Ellipse Volume Formula = \(\frac{4}{3}\) \(\times\) π \(\times\) A \(\times\) B \(\times\) C, where, A, B, and C are the lengths of all three semi-axes of the ellipsoid and the value of π = 3.14 or \(\frac{22}{7}\).
General Equation of an Ellipse
When the center of the ellipse is at the origin (0, 0) and the foci are on the x-axis and y-axis, then we can easily derive the ellipse equation.
The equation of the ellipse is given by: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)
Circumference of Ellipse Formula = π(r1 + r2)
Where, r1 is the semi-major axis of the ellipse and r2 is the semi-minor axis of the ellipse.
Previous Year Questions
- The sum of the focal distances of any point on the conic … [BITSAT 2006]
- The equation … [KEAM 2017]
- A point P moves so that the sum of its distances … [JKCET 2008]
- The eccentricity of the ellipse … [VITEEE 2018]
- An arch of a bridge is semi-elliptical with major axis … [BITSAT 2014]
- Let S and S′ be the foci of an ellipse and B … [TS EAMCET 2017]
- The angle between the lines joining the foci of an ellipse … [WBJEE 2009]
- Eccentricity of ellipse … [BITSAT 2018]
- The distance of a focus of the ellipse … [KCET 2000]
- If one end of a diameter of the ellipse … [KCET 2000]
Things to Remember
- Ellipse is similar to the properties of a circle. But, an ellipse is oval in shape.
- An ellipse is the locus of points in a plane, the sum of the distances from two fixed points (F1 and F2) is a constant value. The two fixed points (F1 and F2) are called the foci of the ellipse.
- Area of the Ellipse Formula = πr1r2
- Perimeter of Ellipse Formula = 2 π√[(r21 + r22)/2]
- Ellipse Volume Formula = \(\frac{4}{3}\) \(\times\) π \(\times\) A \(\times\) B \(\times\) C
- The equation of the ellipse is given by: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)
- Circumference of Ellipse Formula = π(r1 + r2)
Sample Questions
Ques: Find the area of an ellipse whose semi-major axis is 10 cm and semi-minor axis is 5 cm. [2 marks]
Ans: Given,
Semi major axis of the ellipse = r1 = 10 cm
Semi minor axis of the ellipse = r2 = 5 cm
Area of the ellipse
= πr1r2
= (3.14 × 10 × 5) cm2
= 157 cm2
Ques: Find the area and perimeter of an ellipse whose semi-major axis is 10 cm and semi-minor axis is 5 cm. [3 marks]
Ans: Given,
Semi major axis of the ellipse = r1 = 10 cm
Semi minor axis of the ellipse = r2 = 5 cm
Area of the ellipse
= πr1r2
= (3.14 × 10 × 5) cm2
= 157 cm2
Perimeter of the ellipse
= 2π√[(r21 + r22)/2]
= 2π√[(102 + 52)/2] cm
= 2π√[(100 + 25)/2] cm
= 2π√[125/2] cm
= 2 × 3.14 × 7.91 cm
= 49.674 cm
Ques: Find the circumference of ellipse whose semi-major axis is of length 12 units and semi-minor axis is of length 11 units using one of the approximation formulas. Use π = 3.14. [2 marks]
Ans: The length of the semi-major axis is, a = 12 units.
The length of the semi-minor axis is, b = 11 units.
Therefore,
P ≈ π (a + b)
P ≈ 3.14 (12 + 11)
P ≈ 72.22 units
Hence, The approximate value of the circumference of the ellipse = 72.22 units.
Ques: Find the integral used to approximate the perimeter of an ellipse (x2/25) + (y2/16) = 1 and evaluate it using your calculator. [2 marks]
Ans: Comparing (x2/25) + (y2/16) = 1 with (x2/a2) + (y2/b2) = 1, we get
a = 5 and b = 4.
Then eccentricity, e = [√(a2 – b2)] / a = [√(25 - 16)] / 5 = 3/5.
Using one of the formulas for the perimeter of an ellipse using integration,
P = 4a∫O π/2√1 − e2sin2θdθ
P = 4(5)∫O π/2√1 − (3/5)2sin2θdθP
P ≈ 28.3617 units
The perimeter of the given ellipse ≈ 28.3617 units.
Ques: If the length of the semi-major axis is 7cm and the semi-minor axis is 5cm of an ellipse. Find its area. [3 marks]
Ans: Given, length of the semi-major axis, a = 7cm
length of the semi-minor axis, b = 5cm
By the formula for the area of an ellipse,
Area = π x a x b
Area = π x 7 x 5
Area = 35 π
or
Area = 35 x 22/7
Area = 110 cm2
Ques: What are the values of 'a' and 'b' in the equation of ellipse 16x2 + 25y2 = 1600. [2 marks]
Ans: To get the values of 'a' and 'b' we need to write the given equation in standard form
So, divide the given equation of an ellipse 16x2 + 25y2 = 1600 by 1600.
we get x2/100 + y2/64 = 1
So, by comparing x2/100 + y2/64 = 1 with x2/a2 + y2/b2 = 1
The values are a = 10 and b = 8.
Ques: Find the lengths of major and minor axes of the ellipse x2/25 + y2/16 = 1. [2 marks]
Ans: The equation of the ellipse is x2/25 + y2/16 = 1
Comparing the above equation with x2/a2 + y2/b2 = 1,
a2 = 25 and b2 = 16
Length of major axis = 2a = 10
Length of minor axis = 2b = 8
Ques: Find the equation of the ellipse, having a major axis along the x-axis and passing through the points (-3, 1) and (2, -2). [2 marks]
Ans: Since the points (-3, 1) and (2, -2) lie on the ellipse,
x2/a2 + y2/b2 = 1, (-3)2/a2 + 12/b2 = 1, 9/a2 + 1/b2 = 1 and
22/a2 + (-2)2/b2 = 1, 4/a2 + 4/b2 = 1
Solving these equations simultaneously a2= 32/3 and b2=32/5
So, the equation of the ellipse is
x2/(32/3) + y2/(32/5) = 1
i.e., 3x2 + 5y2 = 32
Ques: The length of the semi-major and semi-minor axis of an ellipse is 5 in and 3 in respectively. Find its eccentricity and the length of the latus rectum. [5 marks]
Ans: Given, a = 5 in, and b = 3 in
Now, applying the ellipse formula for eccentricity:
e =√(1 − b2/a2)
= √( 1 - 32 / 52)
= √(1 - 9/25)
= √[(25 - 9)/25]
= √(16/25)
= 4/5
= 0.8
Now, applying the ellipse formula for the latus rectum:
L = 2(b2)/a
= 2(32)/5
= 2(9)/5
= 18/5
= 3.6 cm
Eccentricity and the length of the latus rectum of the ellipse are 0.8 and 3.6 in respectively.
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